4 letters words are to be formed out of the letters of the word PASSPORT Their number is

Options

A. 666
B. 626
C. 686
D. 606

Answer

To find the number of 4-letter words that can be formed from the letters of the word "PASSPORT", we need to consider the letters and their frequencies. The word "PASSPORT" consists of the following letters:

- P: 2 times
- A: 1 time
- S: 2 times
- O: 1 time
- R: 1 time
- T: 1 time

This gives us a total of 8 letters, with the following breakdown:
- Total unique letters: 6 (P, A, S, O, R, T)
- Letters that repeat: P and S (each appears 2 times)

We will consider three cases for forming 4-letter words:

### Case 1: 2 Same Letters and 2 Different Letters
1. **Choose the letter to repeat**: We can choose either P or S to repeat. This gives us \( \binom{2}{1} = 2 \) ways.
2. **Choose 2 different letters from the remaining letters**: After choosing one letter to repeat, we have 5 letters left (since one letter is already chosen). We need to choose 2 from these 5 letters. This can be done in \( \binom{5}{2} \) ways.
3. **Permute the letters**: The arrangement of the letters will be \( \frac{4!}{2!} \) because we have 2 identical letters and 2 different letters.

Putting this all together:
\[ \text{Total for Case 1} = 2 \times \binom{5}{2} \times \frac{4!}{2!} \]
Calculating:
\[ \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 \]
\[ \frac{4!}{2!} = \frac{24}{2} = 12 \]
Thus,
\[ \text{Total for Case 1} = 2 \times 10 \times 12 = 240 \]

### Case 2: All 4 Different Letters
1. **Choose 4 different letters from the 6 unique letters**: This can be done in \( \binom{6}{4} \) ways.
2. **Permute the letters**: The arrangement of the 4 different letters will be \( 4! \).

Putting this together:
\[ \text{Total for Case 2} = \binom{6}{4} \times 4! \]
Calculating:
\[ \binom{6}{4} = \binom{6}{2} = \frac{6 \times 5}{2 \times 1} = 15 \]
\[ 4! = 24 \]
Thus,
\[ \text{Total for Case 2} = 15 \times 24 = 360 \]

### Case 3: 2 Same Letters and 2 Same Letters
1. **Choose the letters to repeat**: The only possible pairs are (P, P) and (S, S). Thus, we can choose 2 letters from {P, S}. There is only 1 way to choose both.
2. **Permute the letters**: The arrangement of the letters will be \( \frac{4!}{2! \times 2!} \) because we have 2 identical letters of one type and 2 identical letters of another type.

Putting this together:
\[ \text{Total for Case 3} = 1 \times \frac{4!}{2! \times 2!} \]
Calculating:
\[ \frac{4!}{2! \times 2!} = \frac{24}{2 \times 2} = 6 \]

### Final Calculation
Now, we sum the totals from all three cases:
\[ \text{Total Words} = \text{Case 1} + \text{Case 2} + \text{Case 3} = 240 + 360 + 6 = 606 \]

Thus, the total number of 4-letter words that can be formed from the letters of the word "PASSPORT" is **606**.